IEEE TRANSACTIONS ON MICROWAVE THEORY AND ‘TECHNIQUES, VOL. MTT-14, NO. 12, DECEMBER, 19.s6 635

UNCOMPENSATEDSAMPLER

E

‘b %3o 124GHz o f+ 12.4GHz

BANDWIDTH BANDWIDTH

,i~ ~

v

;R

1.7:I --—> --_—-—

f+ 12.4GHZ 0 f+ 124GHz

Fig. 15. .4verage measured performance.

CONCLUSION

The bandwidths obtained, using the electromechani-

cal configuration just described, are well in excess of

1~.4 GHz. This represents an increalsein bandwidth of

three to four times that previously available. It is safe to

assume that this technique will eventually be used to

achieve bandwidths in excess of 18 (GHz.

ACKNOWLEDGMENT

The author wishes to acknowledge the helpful sugges-

tions of Dr. B. Oliver, Vice President of Research at

the Hewlett-Packard Company, and the devoted work

of the many individuals at -hp- Associates, who con-

tributed to the development work described here.

J?EFERENCES

[1] R. Sugarman, ‘[Sanlpling oscilloscope for statistically varyingpulses, ” Rev. SCi. Inst~., vol. 28, pp. 933-938, November 1957.

[2] J. M. L. Janssen, “An experimental ‘stroboscopic’ oscilloscopefor frequencies up to about 50 me/s, ” Phillips Tech. Rev., vol.12, pp. 52–58, 1950 and vol. 12, pp. 73–81, September 1950.

[3] E. Hospitalier, “The slow registration of rapid phenomena bystrobographic methods, ” The Elec. En.gr. (Melbotime, A ustraliu),pp. 4&44, January 1, 1904.

[4] G. B. B. Chaplin “A method of designing transistor avalanchecircuits with application to a sensitive transito,r oscilloscope, ”Dipest of Tech. Pabers. IRE-AIEE 1958 Transistor am! Solid-

[5]

[6]

[7]

[8]

[9]

[10]

[11]

St;te C&uits Conf.: PP; 21-23.H. L. Callendar, “An alternating cycle-curve recorder, ” TheElect~ician, pp. 582–586, August 26, 1898.E. W, Gelding, Electrical Mease.wentents and ilfeaswing .lnstr~d-nzents, 3rd cd., revised. London: Pitman and Sonq, 1942.R. J. D. Reeves, “The recording and Collocation of Wave-forms, ” Electronic Etzgrg., vol. 31, pp. 130–137, March 1959 andvol. 31, pp. 20&212, April 1959.J. G. McQueen, “The monitoring of high-speed wave Forms, ”Electronic Engrg., pp. 436441, October 1952.F. A. Laws, Elect~ical Measurements, 2nd ed. New York: Mc-Graw-Hill, 1938.R. Carlson et al., “Sampling oscillography, ” IRE WESCONRec., pt. 8, pp. 44-51, 19.59.R. Carlson, “A versatile new DC-5OO MC oscilloscope with highsensitivity and dual channel disdav, ” Hewlett-.Packard .T., vol.

[12]

[13]

11, nos. 5-7, January/March 1960. -G. Frye and N. Nahman, “Random sampling oscillography, ”IEEE Trans. on Instrumentation and Measurement, vol. [M-1.?,pp. 8–13, March 1964.C. Yen, “Phase-locked sampling instruments, ” IEEE Tram. onInstmmentation and Measurement, vol. I M. 14, pp. 6&68,hIarch/Iune 1965.

[14] W. Gro~e, “A new DC-400 MC sampling ‘scope plug-in withsignal feed-through capability, ” HezvZett-Packard -T., vol. 15, no.8, pp. 5-8, April 1964.

[15] H. ‘Wallman and G. E. Valley, Jr,,New York: McGraw-Hill, 1948.

[16] J. Truxal, Control System Synthesis.1955, pp. 36-40.

[17] .Sb$~ Schelkunoff, Advanced A ntenna

Vacuwn Tube Am@ijiers.

New York: McGraw-Hill,

Theory. New York: Wiley,

Theoretical and Practical Applications of Capacitance

Matrix Transformations to TEM Network Design

R. J. WENZEL, MEMBER, IEEE

Ahstracf-TEM propagation on an array of parallel conductors

is described in terms of the normalized static capacitance matrix.

Important properties of capacitance matrices are discussed and a

physical and network interpretation is given to a useful linear trans-

formation of the static capacitance matrix. Several practical applica-

tions of capacitance matrix transformations are given. These include

Manuscript received June 17, 1966; revised A~gust 2, 1966.This work was supported by the U. S. Army Electromcs Laboratory,Fort Monrnouth, N. J., under Contracts DA 28-043 AMC-00399(E)and DA 28-043 AMC-01869(E).

The author is with The Bendix Corporation, Research Labora-tories Division, South field, Mich.

1) equivalent circuits for directional couplers with equal tsmnina-

tions, 2) design procedures for directional couplers with unequal

terminations, and 3) element value tables and design details for

compact coaxial filter-transformers. Construction details and experi-

mental results are presented for a 3:1 bandwidth fiker-transformer

constructed with multiple re-entrant coaxial lines.

AN

1. INTRODUCTION

IMPORTANT ASPECT of TEl\I quarter-

wave network synthesis and design is the multi-

plicity of physical configurations that yield iden-

636 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

tical response characteristics. Different network con-

figurations are often required to realize a desired re-

sponse for moderate changes in design parameters be-

cause of the small range of realizable impedance values.

In most design procedures (either exact or approxi-

mate), practical circuit element values are obtained by

application of suitable equivalent circuit relationships.

In a previous paper [1], the concept of obtaining in-

variant transmission characteristics using transforma-

tions of the static capacitance matrix of a parallel-

coupled line array was introduced and was applied to

the exact design of interdigital bandpass filters. The

purpose of this paper is to demonstrate in more detail

the importance of the static capacitance matrix, to

show how numerous equivalent circuits can be derived

by its transformation, and to give a simple network and

physical interpretation to the transformation.

The body of this paper is presented in Sections II and

II 1. In Section II-A, the description of TEM trans-

mission on an array of parallel conductors is given in

terms of the static capacitance matrix and some basic

properties of capacitance matrices are described and

related to specific network geometries. In Section II-B,

the capacitance matrix transformation is defined and a

network and physical interpretation of this transforma-

tion is presented.

The application of the capacitance matrix transforma-

tion to the design of directional couplers with equal or

unequal terminations is presented in Sections II I-A and

III-B. Element value tables and design information

relative to the construction of compact filter-transform-

ers based on the interdigital filter prototype are pre-

sented in Section III-C. Measured performance char-

acteristics for an experimental 3:1 bandwidth filter-

transformer are also given.

11. TENI MODE COUPLED LINE THEORY AND THE

CAPACITANCE n~ATRIX TRANSFORMATION

A. The Static Capacitance Matnk C

TEN I propagation on an array of N+ 1 parallel con-

ductors can be described in terms of an NX N static

capacitance matrix by consideration of the 2iV coupled

partial differential equations that apply to such a sys-

tem. This problem has been considered by several au-

thors [~]– [4] and it is readily shown that the 2.Wport

voltages and currents defined in Fig. 1 are related by-

the admittance matrix Y as follows:

[1I.i [

c1

/ . c~~.~

—

1 1

———————&——————? 0s

~B –C<l –s’ / c

1- 1

v.]., ––– , (1)

I,A _

‘IA t

E-

t v, B””— 2N PORT =

A-ENO . ARRAY OF . B-. N + ? PARALLEL _

COUPLEOINA_ ● LINES

● —INB

‘NAt f v~B

— —

DECEMBER

ENO

Fig. 1. 2N-port network with admittance matrix Y.

C = NXN normalized static capacitance matrix with

elements c,

~ = CI/e = ratio of the static capacitance between con-

ductors per unit length to the permittivity of the

medium (this ratio is independent of the dielec-

tric medium and depends only on cross-section

geometry). This dimensionless ratio is convenient

for practical use with available design data [5],

[6].

vo = 376.6 ohms, the impedance of free space,

S=j tan e

6 =7rco/2GJ0 = electrical length of the lines in radians,

and

OJO= the frequency for which the lines are a quarter

wavelength.

Other network matrices (Z, ABCD, S, etc. ) can be ob-

tained by well-kno~rn transformations of ( 1) (see Refer-

ence [7]).

Some general properties of capacitance matrices will

be discussed by considering the arbitrary array of four

conductors shown in Fig. 2(a). Conductor 4 is taken as

reference (ground). The 3 by 3 capacitance matrix has

main diagonal terms given by the sum of all capacitors

connected to each of the three ungrounded conductors

and off-diagonal terms given by the negative of the

capacitors bet~~-een conductors. Therefore

[

( I

c1 + C12 + CIJ I —C12 I —C13

I 1I ––––––l––––––l–––––– I

I Ic= —C12 I C2+ c~~ + C?J [ —C23 (2)

I Ii ——————l—————— l—————— I

1 I I—C13 — C23 \ C3 + C13 + CZ3

I I 1

The elements of matrix (2) satisfy the following rela-

tionships [8], [9]:

d) c{~ s O i#k. (3)

1966 WENZEL: CAPACITANCE MATRIX TRANSFORMATIONS 637

&

(0) Gener.l Arroyof Par.llel Coupled Line.

&C12 ‘ 23

7

1pz ~’23 ~c3

(b) Paro[le lCoopled Array W,ih COUpl,ng,I.aNearest Neighbor, Only

Fig. 2. Static capacitances of parallel collpled line arrays.

To obtain a physical realization for a network of the

type shown in Fig. 2(a), starting from a given matrix

C, requires that physical dimensions be determined

either experimentally or analytically by solution of

Laplace’s equation. Unfortunately, this is an extremely

difficult problem, for most geometric configurations and

solutions are known only for a few special cases. One

case of particular interest is obtained by assuming that

each conductor is coupled directly only to its nearest

neighbors. This configuration is shown in Fig. 2(b),

where conductor 4 is assumed to be a ground plane. The

capacitance matrix for this network is given by

c=

I Ic1 + f-12 I — c~~ 10

———=

—612

1--”0

___. _.– —______t

CIZ + C2 + C23 I ‘CZ3

—–———–l-–——–—C23 I c3 + 623

I

(4)

and allow considerable versatility of cross-section geon~-

etry, as will be shown in succeeding sections. Through-

out the remainder of the paper, only network geometries

with lines directly coupled to nearest neighbors will be

considered.

B. The Capacitance Matrix Transformation

Different network configurations that leave some

specified network response invariant are said tcl be

“equivalent.” A method of investigating possible equiv-

alent circuits is to use a linear transformation of net-

work variables (voltage and/or current, for example).

Linear transformations of network variabIes can be lper-

formed in a manner such that network realizability is

assured (at least from a theoretical viewpoint). H[ow-

ever, any such transformation will be of p12LCtiCd util-

ity only if it is easily interpreted and/or leads to a de-

sirable physical structure.

Guillemin [8] has shown that a useful linear trans-

formation is obtained by multiplying the rows and col-

umns of a parameter matrix by suitable constants. In

carrying out this transformation on a static capacitance

lmatrix, symmetry requires that each operation be ap-

plied to both a row and its corresponding column.

hIathematically, the transformation is given by

‘?ll -

‘n2-

C’= ~JJcnN+

i-

—Clz

o

I

I

Czp

_——

I

(5)

where n~- is a diagonal matrix with elements z1, 732, . . . ,

The C matrix for a like array with any number oi con-

n+v. The transformed capacitance matrix C’ is obtained

by pre- and post-multiplication of a given matrix C by

the diagonal matrix nN. The right-hand portion of (5)

indicates a convenient notation for use in applying the

ductors is obtained in a similar manner. Note that the

assumption of coupling only to nearest neighbors results

in a matrix with nonzero elements on the main diagonal

and adjacent to the main diagonal only. C~iven a C

matrix of the above t~-pe, data exist from which physi-

cal dimensions can be determined [5], [6]. It is empha-

sized that the assumption of coupling to nearest neigh-

bors only is not necessarily desirable from the vieu--

point of obtaining optimum device performance from

a given number of lines. ILlany useful devices might

result if simple solutions to Laplace’s equation were

known for a larger number of geometries. On the other

hand, networks with lines coupled only to nearest

neighbors can realize a wide class of network functions

transformation to practical problems and serves t:o il-

lustrate how each ith row and its corresponding column

is multiplied by the same constant ni.

To give a network interpretation to the capacitance

matrix transformation of (5), consider the additicm of

ideal transformers of turns ratio n,: 1 to the 2N-ports

of the .Wline network sho~vn in Fig. 3(a). The resultant

network is shown in Fig. 3(b). Physically, a new net-

work has been formed with voltage and current vari-

ables related to those of the original network by

1 This section is directed toward the definition and interpretationof capacitance matrix transformations. A more general discussion~~le_qlu~-&ent network concepts can be found in Gui Ilemin [8], pp.

638

[ VA’

——

1VB’

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

where nN is a diagonal NXN matrix with elements

%1,%2,..., n,v, and n,v–l is a diagonal NXN matrix

with elements l/nl, l/nz, “ “ . , l/nN. Then from (1)

and (6)

[

lA’ 11

——— .

1~’70S

1- -1

where

Thus

r

In~r I o 1[ c – cdl – s’

1[;1

nNIo__ —/———

; 1

———~_——

0 I n.v –C<l – s’ c o I nN

[’1–1

nN–l I o——–~—–— .

0 / n~–l J

1.4’ 1[ nNCn,v11

I = —1 —————————

InNIO ]

1–––~–––l

o I nh,

I 1

—————————

nNcn.V

r c’I

1 I——

m.s1

–Cf<l – S2 ~

I

VA’

[

VA’———

VB’

DECEMBER

(6)

VA’

I (7)

VB’

1

1“ (8)

Referring to (8), the new matrix C’= nNCn,v is

exactly that given by the capacitance matrix trans-

formation of (5). However, t] \e new matrix C’ is also a

matrix corresponding to a parallel line array and can be

realized by simply changing the cross-sectional geom-

etry from that of the original network as depicted in

Fig. 3(c). The following statement summarizes the net-

work and physical implications of the capacitance

matrix transformation:

given a parallel line array, transformation of the cor-

responding matrix C results in a matrix C’ that cor-

responds to a new network identical to the original

with ideal transformers of turns ratio n,: 1 added at

both ends of each line. Physically, the new network

can be realized by a change in cross-sectional geom-

etry.

The factors ni used in performing a capacitance matrix

transformation can be chosen arbitrarily as long as they

lead to a transformed capacitance matrix whose ele-

ments obey the restrictions given by (3).

The remaining question to be investigated is the fol-

lowing: under what conditions can the cross-sectional

geometry be altered while a specified network response

is held invariant? The answer becomes clear upon study-

ing the diagrams of Fig. 3. If each of the 2N-ports of the

parallel line array of Fig. 3(a) is terminated in a passive

admittance Y. (i= 1 . . . 2N) then addition of ideal

transformers [as in Fig. 3(b)] will affect only the net-

work admittance level at any driven port. However,

each terminating admittance Y~ can be transformed

through the ideal transformers to yield primed termi-

nating admittance Y, ‘ = n,s Yi. The original network and

ideal transformers can then be replaced by a new net-

work with a different cross-sectional geometry as de-

scribed. The above statements are summarized as follows:

the transmission response of a parallel line network

with capacitance matrix C and line terminations Yi

is identical to that of a parallel line network with

capacitance matrix C’ and terminations Yi’ = niz Yi,

where the fii are the constants utilized in transformi-

ng C into C’.

1966 WENZEL: CAPACITANCE MATRIX TRANSFORMATIONS 639

(a) NetWork W,th C.p.., !.n.e Metr,. C

(b) NelWork W,th C.pocito”ce Motr,x C(At ?rsrnedTemin.ls)

(C) Network W,th CoP.c, t.nce M.tr, xC’

Obto,ned by Chonq,ng CrossSecttonol Geometry

Fig. 3. Network s.nd physical interpretation of thecapacitance rnatrixtransformation.

v,Bf1V2B

4P--(.) Meonder Line Configuration (b) Effect Of Copo.it.nce M.trix

Trmsfornmt ion

Fig. 4. Example illustrating inadmissible terminating conditions.

From the above statement, all cc,mplex terminating

admittances, including opens and shorts, can be accom-

modated. I t is emphasized that ordy terminations con-

nected at the ports defined between the end of each line

and the ground plane be considered in obtaining equiv-

alent circuits by a change in cross-sectional geometry.

An example of a network configuration for which an

equivalent circuit cannot be realized by a change in

cross-sectional geometry only is the familiar meander

line shown in Fig. 4(a). Only two lines are shown for

simplicity. Transformations of the capacitance matrix

of the basic array introduce ideal transformers as shown

in Fig. 4(b). As is readily seen, the original boundary

condition that VIB = VzB cannot be satisfied by making

V’lB = V’ZB except when nl = n2. However, choosing

ml= nz is a trivial transformation that merely changes

the impedance level of the entire network. The above

example does not invalidate the use of the capacitance

matrix transformation, but emphasizes that only under

the specified terminating conditions can the transforma-

tion be interpreted as a simple change in cross-sectional

geometry.

II 1. TEM NETWORK APPLICATIONS OF CAPACITANCE

lL’lATRIX TRANSFORMATIONS

A. Symmetric Directional Coupler Equivalent Circuits

As an example of some familiar equivalent circuits

obtainable by application of the capacitance matrix

transformation, consider the matched single-section

four-port parallel line directional coupler shown in Fig.

5(a). The matched constraint requires that Z.= ti~~~~Z~~,

where Zo. and Zoo are the even- and odd-mode imped-

ances and Zo is the characteristic impedance of the

terminating lines [10]. The normalized [1] even- and

odd-mode static capacitance values (c) are related to the

even- and odd-mode impedances [10] by

Zo, = -&--- and Z~~ == ~—– “ (9),“ d c, . coo

The corresponding self and mutual static capacitance

values are then those given in Fig. 5(b). The circuit of

Fig, 5(b) can be considered as a degenerate three-line

network above a grounded plane in which the center

line is coupled to both outer lines, but is decoupled from

640 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES DECEMBER

&e(a)

(b)

‘m -’.. ‘.. -c.,

P----- ‘--?‘“”T T<”’

&

(c)

(d)

c.. .0.

‘Y.opT-

(f)

Fig. 5. Directional coupler equivalent circuits obtained by capacitance matrix transformations.

ground as shown in Fig. 5(c), The electrical perfor- transformed network is seen to be the re-entrant section

mance of this network at ports one through four is iden- described by Cohn [11]. The two-line directional

tical to that of the two-line network of Fig. 5(b). The coupler and the re-entrant coupled section are limiting

capacitance matrix for the network of Fig. 5(c) is cases of a more general three-line network. The two-line

[

Icoo I — (coo — co.) 1 0

——–—–—/—–—––--/–—–—–—

I – (coo – co.) I ~ (coo – co.) i – (coo – co.)

1

——————/——————/——————

o I — (c.. — c.e) 1 coo

T’‘u

network results from a transformation that decouples

the center line from the ground plane and the re-entrant

section from a transformation that decouples the outer

lines from the ground plane.- n. (lo)

B. Directional Couplers with Unequal Terminations

The capacitance matrix transformation allows the

simple design of directional couplers with unequal

terminations. To demonstrate the technique, consider

again the matched four-port directional coupler with

If the center row and column of this matrix is multiplied

by an admissible constant n, the electrical performance

at ports one through four is unchanged. The effect of a

partial capacitance matrix transformation on the cor-

responding physical realization is shown in Fig. 5(d).

A complete transformation results when the multiply-

ing constant is chosen such that the outer lines are de-

coupled from the ground plane. This requires that the

sum of elements in the first row or column of the

capacitance matrix be zero. Therefore,

Go – n’(coo – co.) = O or n’ = co” . (11)coo — Coe

The completely transformed netxvork capacitance values

are shown in Fig. 5(e) and a physical realization with

appropriate impedance values is shown in Fig. 5(f). The

coupling coefficient k shown in Fig. 6(a). The corre-

sponding capacitance diagram is show-n in Fig-. 6(b)

where the capacitance values are those given in Section

III-A. The coupling coefficient is given by

*(COO – cod) coo — C“ek=

*(COO– LX?) + co. = c.. + c“. “(12)

The appropriate capacitance matrix is

.[:1co. + co, I Coe— coo1

____/ —— __ .

T“(13)

co. – coo I coo + co.. . * u

‘r

If an asymmetric transformation is performed by

1966 WENZEL: CAPAClTANCE MATRIX TRANSFORMATIONS 641

(a)

cOet=T-L

.

(b)4’

1d.2

n:

+ (Coo +coe)

Q-;(coo

1 .2

—

2 coo . . . (.OO + @

(COO-%,)2

+

(c)

‘0 e

4’

,$$

3;, :2

% J 2 coo . . . (coo + co,)

1 2’ /(.OO- ..,)2

==

1 !?

(d)

Fig. 6. Directional couplers w-ith unequal terminations.

multiplying the second row and column by a factor W, a

new: capacitance matrix is obtained IV hose network real-

ization is the original network with ideal transformers

of turns ratio n: 1 added to both ends of the second line.

As described in Section II-B, if the unit terminating

conductance are changed to a value of n~, the original

network with the ideal transformers added can be real-

ized by a change in cross-sectional geometry while

maintaining the transmission properties invariant. The

network interpretation and a physical realization of a

coupler corresponding to a partial transformation are

sho~vn in Fig. 6(c).

.4 complete transformation results if the self-capaci-

tance from the first node is eliminated. From (13) this

requires

Ih (600— Coe) = L-Oo + cog

or

coo + coo 1jl = =—. (14)

&l — Go, k

A coaxial physical realization of a coupler correspond-

ing to a complete capacitance matrix transformation is

shown in Fig. 6(d) along with the appropriate capaci-

tance network. From (14), the maximum ki7TTkTatiOTT

ratio is seen to be dependent on coupling level, the ratio

being given by

1‘rlm,x 2 =_.

k2

As an example, a 3-dB coupler can yield at most a

1 1=2:1

;’ (0.707)’

(15)

termination ratio. Also, as is evident from (15), the

lower the coupling coefficient, the higher the possible

termination ratio. The above procedure yields a transf-

ormation from a unity admittance value to a larger

admittance value. Transformations from unity to

smaller admittance values are obtained by changing the

admittance level of the entire network ancl using the

transformed line as reference.

C. Compact Filte~- Transformers

In a previous paper [1], symmetric capacitance nla-

trix transformations were utilized to obtain partial or

entire coaxial realizations of interdigital filters. By ap-

plying an asymmetric transformation, compact coaxial

interdigital filters that impedance transform in the pass-

band, can be designed. These designs use the element

value tables developed for the exact design of inter-

digital filters [12 ]. As with all impedance transforming

devices, there are constraints imposed on the impedance

ratio that can be achieved over a given bancl~vidth with

a given tolerance and number of sections. These con-

straints have been investigated by straightforward ap-

plication of the capacitance matrix transforrnation. The

new network forms were found to be comparable in inl-

pedance transforming properties to quarter-viave

stepped impedance designs using a like number of sec-

tions.

The normalized self and mutual static capacitance

element values for interdigital filters with open-circuited

terminating lines (OCTL) are given in Tables I–II I for

four, five, and six section designs, respectively.2 ‘Two

ripple values, 0.01 dB (1.10 VSWR maximum) and 0.10

dB (1.36:1 VSWR maximum) are given. The tabulated

element values can be used directly with available data,

[1], [5], [6], and [13] to obtain physical dimensions for

interdigital filters.

Z More extensive element value tables including networks withshort-circuited terminating lines (SCTL) are given in Reference [12].The OCTL networks give a higher termination ratio for a specifiednumber of sections and passband ripple than do the SCTL networks.The three tables given should be adequate for most filter-transformerrequirements,

642 IEEE TRANSACTIONS ON MICROW ‘AVE THEORY AND TECHNIQUES

The following symbols are used in the tables pre-

sented:

Nf = number of filter sections (4, 5, or 6)

m = 3, number of L-C elements in basic proto-

type3

n = number of unit elements in basic proto-

type (1, 2, or 3)3

~=ripple value in dB (0.01 dB and 0.10 dB)

VSViH2.W = maximum theoretical standing-wave ratio

in the filter passband

% BW = percentage bandwidth

FH/F~= (200+~o BW)/(200–~o BW) =ratio of

high passband edge frequency to low pass-

band edge frequency

R8/R~ = maximum termination ratio determined by

analysis. General formulas are given in

each table for the specific configuration

shown.

Concerning the filtering properties of the filter-trans-

formers, the stop-band attenuation at transformed fre-

in Fig. 7(a). The capacitance matrix is:

411.0 –11.0 o

1 –11.0 23.58 –7.19 o

d; o

1

–7.19 23.58 –11.0 “(17)

o –11.0 11.0Lo

01

The network of Fig. 7(a) is symmetric and is designed

to work between equal terminating impedances. Un-

equal terminating impedances can be obtained by an

asymmetrical transformation of the capacitance matrix,

as discussed in Section II.

A desirable network configuration is one that con-

tains only one shunt c in the corresponding capacitance

network. This allows all other elements to be incor-

porated coaxially in the element represented by the

single shunt c.

Referring to matrix (18), a single shunt c can be

achieved by choosing nl and nz such that the desired

network configuration, as shown in Fig. 7(b), is ob-

quency fl~ is given by [12], [14]; tained.4

r 11.0 -11.0 0 0 -

1 I –11.0 23.58 –7.19ZI o

‘147 0 –7.19nl 23. 58n12 – 11 .Onlnz

o 0 –ll. OnlnZ 11.0na2 .

‘r ‘rnl 922

[ (A = 10 log10 1 + C2cosh2 m cosh–l ~

.—<1 + Q.’

+ n cosh-’/1 + Q.’ )1(16)

where

A = attenuation in dB

C2=(10’11O—1)

0.= tan 7rj0/2f0

!d~ = tan ~fA/2fofo= ~~;;cY at which the lines are a quarter wave-

fa=cutoff frequency =fO[l– %BW/200]

f*= frequency at which the attenuation is A dB

To demonstrate the use of the tables and the design

techniques involved, a 3:1 bandwidth four-section filter

transformer with 0.01 dB passband ripple will be de-

scribed. The prototype self and mutual static capaci-

tance values are obtained from Table I-A and are shown

a The significance of m and n is described in detail in References[12] and [14].

+-7’21

+- 722

(18)

This requires that:

–11.0 + 23.58 – 7.19nl = O

ll.OnlfZZ = 11.0n22

- nl = nz = 1.75.. . (19)

Then Rs/R~ = termination ratio= nz2 = 3.06.

Multiplication of the outer row and column of matrix

(17) by nl = 1.75, as indicated in matrix (18), lowers the

impedance level by nz2 = 3.06, and requires the use of a

l/n22 = 0.327 ohm load5 to maintain a 0.01 dB ripple

response in the passband. The transformed network

thus acts as a 3.06:1 impedance transformer. The im-

pedance ratio that can be achieved is limited by the

choice of nl and nz, which are constrained, as described

in Section II. A higher impedance ratio can be achieved

by applying another transformation such that the single

shunt capacitor is connected to the fourth node. Re-

ferring to matrix (18), this requires that:

4 Note that the formulas for R,s/RL given in each table allow thecalculation of the achievable termination ratio for the configurationsshown. Actual capacitance values are obtained as described in thedesign example.

5 @ a normalized l-ohm basis.

1966 WENZEL: CAPACITANCE MATRIX TRANSFORMATIONS 643

–11.0 + 23.58 – 7.19nl = O

– 7.19nl + 23,58n12 – 11. On,n, = O. (20)

Solving gives

nl = 1.7.$

+L* = 3.09, (21)

and results in a termination ratio 01 nzz = 9.6:1. The

corresponding capacitance network is shown in Fig.

7(c). Whereas the network of Fig. 7(c) is capable of pro-

viding a higher impedance ratio over al given band than

the network of Fig. 7(b), all elements must be incor-

porated coaxially in the end element. This requires the

end element to be large and results in a difficult con-

nector problem. For this reason a network form such as

that of Fig. 7(b) is often preferable. Furthermore, in

some applications it is desirable to have not only a low

impedance but also to have a region of high current

density. This can be achieved by using a configuration

such as that of Fig. 7(b).

Before describing experimental results, it is inter-

esting to compare the transformer prc)perties of the co-

axial filter-transformer with those of quarter-wave

stepped impedance lines. From l“oung’s Tables [15], for

3:1 bandwidth, four quarter-wave sections give a maxi-

mum VSWR of 1.07:1 for a 3:1 impedance ratio and a

maximum VSWR of 1.18:1 for a 10:1 impedance ratio.

The partially transformed filter network [Fig. 7(b)] is

comparable to the four-section quarter-wave stepped

impedance design and the fully transformed network is

superior to a stepped impedance design.

A trial transformer was constructed based on the

capacitance network shown in Fig. 7(b). The correspond-

ing .S’-plane equivalent circuit on a 50-ohm basis is

shown in Fig. 8(a). The element values (CI, Z, L, Cz) are

directly related to the static capacitor values, as shown

in (22). The corresponding impedance values on a 50-

ohm basis for all-coaxial realization are:

1 no 376.7=21= —– —---=

c35.2 ohms

dzco – 11.0

376.7z=z2=~=——-= 30.0 ohms

~z C, 12.58

?’0 376.7L= Zx=— — = 14.6 ohms

~TT = 35.8

1 ~o 376.7=z4=— — = 11.2 ohms

z 47; = 33.7

(22)

where

CO– Cb= static capacitance values shown in Fig. 7(b).

The corresponding diameter ratios are [13 ] (for teflon,

dz=l.44)

dzl = 2.28in t,eflon

dz2 = 1.65i11 air

dZ3 = 1.27ia air

dzd = 1.34in ~eflOn. (23)

A schematic cross-sectional drawing showing the ltJca-

tion of the various elements and appropriate impedances

is shown in Fig. 8(b). Symbols have been added to im-

portant junction points in Figs. 8(a) and 8(b) to aid in

visualizing the relationship between the prototype net-

work and its physical realization. A detailed line draw-

ing of the filter assembly, including dimensions, is given

in Fig. 9.

The design center frequency chosen was 2. CIGHz. The

line elements were arbitrarily shortened by 0.100 ‘inch

to compensate for end effects. Since a 50/3.06= 116.3

ohm load was not available, a 50-ohm connector \vas

attached to the low impedance end and tests were ,con -

ducted by measuring the VSWR and insertion loss (on a

50-ohm basis. The input impedance of a doubly termi-

nated Chebyshev filter with small passband ripple is

essentially real in its passband; therefore, a good indica-

tion of trial design performance can be obtained by

measuring the input VSWR at the low impedance lend.

The VSWR value should be very nearly the transformer

impedance ratio of 3.06:1. Again, because of the essen-

tially real character of the input impedance, the per-

formance that would have been achieved if a 16,3 ohm

load had been available can be computed to good ap-

proximation by

VSWR511 h,. I..dVSWR16.4 ohm load =

3.06 ‘

when VS WRSO ohm load >3.06 (24)

3.06VSWR16.4 ohm Ixvi = —

VSWR50 ohm load ‘

<3.06,when VSWRSO Ohm load _

The measured and computed performance character-

istics, together with theoretical values, are shown in Fig.

10. The experimental characteristics shown were

achieved with the initial design—no alterations in di-

mensions were made. The performance is observed to be

excellent in the initial design and indicates that trans-

formers of the type described offer the possibility of

achieving wideband impedance transforming networks

in a very compact configuration. A photograph of the

disassembled filter-transformer is shown in Fig. 11.

The filter-transformers described need not use an all-

coaxial realization. Partial capacitance matrix trans-

formations can be performed that allow coupled bar

realizations to be employed, as described in [1].

Various geometric configurations can be investigated

using the capacitance matrix transformation to achieve

a desired configuration and/or more convenient line

impedance values.

644 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

TABLE I

SE~E AND MUTUAL STATIC CAPACITORS EOR CHEBYSHEV INTERDIGITAL FILTERS WITHOPEN-CIRCUITED TERMINATING LINES m =3, n= 1, Nf =4 SECTIONS

A

I. . 0.01 DB VSWRI:= 1.10:1

%,. A’, K’, /?’,

30,0 ,255E+01 .316E+02 .719E+0135, o ,300E+01 .262E+02 .719E+0140,0 ,346 E+01 ,221 E+02 .719E+0145,0 ,393E+01 ,190 E+02 .719E+0150,0 ,443E+01 .165E+02 .719E+0155, o .494 E*01 .144E+02 .719E+0160.0 ,547E*01 .127E+02 .719E+0165.0 ,602E+01 ,11 SE+02 ,719E+0170.0 ,661E*01 .101E+o2 .719E*0175,0 ,722 E*01 .902E+01 ,719E*0180.0 ,788E*01 .811E+01 .719E+0185,0 ,857E+o I ,730E+01 .719E+0190.0 ,932 E+01 .659E+01 .719E+Oi95.0 ,101 E+O2 .596E+01 .719E*01

100.0 ,11 OE+O2 .539E+01 .719E*01110.0 ,129 E+02 .441 E+01 .719E+01120,0 ,153 E+02 .360 E+01 .719E+01130,0 .183E+02 ,292E+01 ,719E+01140.0 ,222E+02 .233E+01 .719E+01150.0 ,276E+02 ,183E+01 .719E+01

.rH

..,,

B

r = O.1ODB VSWRM= 1.36:1

y,,. <’0 4’, 4’,

30,0 ,164E+01 ,354E+02 .648E+0135.0 ,192E+OI ,295E+02 .648E+0140.0 ,222E+01 ,250 E+02 .648E+0145.0 .252 E+01 .215E+02 .648E+o I50.0 ,283E+01 ,188E+02 .648E+0155, o ,315E+01 .16 EJE+02 .648E*0160.0 ,349E+01 ,146E+02 ,648E*0165. o .384 E+01 ,130 E+02 .646E*01

70.0 ,42iE+Oi ,116E+02 .648E+0175.0 ,459E+01 .105E+o2 .64 BE+0180.0 ,500 E+OI ,942 E+01 .648E+01E15. o ,544 E+01 ,850 E+01 .648E+0190,0 ,590 E+01 ,769E+01 .648E+0195.0 ,640 E+OI ,697E+01 .648E+01

100.0 ,693E+01 ,631 E+01 .648E+01110.0 ,815E+01 .519E+01 .648E+01120.0 ,962E+01 .425E+01 .648E+01130,0 .115E+02 .345E+01 .648E+01140.0 ,13il E*02 ,276E+01 .648E+01150,0 ,171E*02 .216E+01 ,648E+01

m m!,,,,~’ 200- Z BW

R=lml~lmRL

R’ [12

FL. l+H

‘2

TABLE 11

SELF AND MUTUAL STATIC CAPACITORS FOR CHEBYSHEV INTERDIGITAL FILTERS WITH OPEiWCIRCUITEDTERMINATIiWG LINES m =3, n =2, Nj = 5 SECTIONS

A

r = 0.01 DB VSWR\f= 1. 10,1

%33V; <’(l e’, ~’, fi’3

30,0 ,240E+oi .350E+02 .688E+oi .?93E+02350 O .282E+oi ,29i E+02 .688E+Oi .235E+0240.0 .326 E+01 .246E+02 .689E+01 .193E*0245.0 ,37i E+oi .212E+02 .690E+oi .160 E+0’2

50.0 ,4i7E+oi ,i84E+02 .691E+01 ,134E+0255. o ,465 E+Oi .162E+02 .692E+01 .li3E+0260.0 ,5i5E+Oi .i43E+02 .693E+ol .960 E+oi65.0 ,567E+oi .i27E+02 .694 E+oi ,8i9E+oi70.0 ,622E*01 .ii3E+02 ,695E+oi ,701E+oi75,0 .681E+Oi .101E+O2 .697E+01 .60i E+oi80.0 ,742 E+01 .910 E+O1 .698E*01 .515E+0185.0 .808E*01 .820 E+oi .699E+oi ,442E+ol90.0 ,879E+01 .739E+01 .70i E+Oi .379E+0195.0 .955 E*Oi ,667E+oi .702E+ol ,325E+oi

100,0 .104 E+O2 .602E+oi .703 E+oi .P78E*oi110.0 ,123E+02 . 49i E+oi .706 E+OI .20i E+oi120.0 .145E+02 .398E+01 .708E+oi .143E+01130.0 .174E+02 .320 E+01 .7io E+oi .994E+o0140,0 ,212E+02 .254E+oi .7i3E+oi .665E+OO

i50,0 ,265E+02 .i96E+Oi .714E+01 .423E+o0

200

B

r = O.1ODB VSWR\f= l,3t:l

30.0

35.0

40.045.050.055,0

60.065, o

70.075.0

80,0

85.090,0

95.0100.0110,0120.0

130.0

140.0

150.0

,i58E+Ol .376 E+02 .629 E+01 ,322 E+02,i86E+Oi ,313E+02 .630 E+01 ,261E+02,214E+01 .266E+02 .630 E+01 ,2i5E+02,243E+01 ,229E+02 .630 E+01 .180 E+02.274 E+01 .200E+02 .631E+oi ,152E*02,305 E+01 .176E+02 ,63i E+oi ,129E+02

.337E+01 .156E+02 .632E+oi ,111 E+02

,371 E+Oi .i39E+02 .632E+01 ,955 E+01,407E+oi .124E+02 .633E+01 ,E25E+oi,444E+01 .ii2E+02 .634E+01 .714E+01

,484E+oi .101E+o2 .634E+oi .6i8E+o I,526E+oi .91oE+O1 .635E+01 ,536E+01,571E+Oi .82SE+Oi .636E+oi ,464E+oi

,619E+01 .745E+01 .636E*01 ,402E+01.671E+oI .675E+01 .637E+01 ,347E+oi

,789E+01 .553E+01 .639E+oi .257E+ol,933E+OI .45iE+oi .640E+oi ,i87E+oi

.lilE+02 ,365E+01 .642E*01 .132E+01

,135E+02 .29iE+Oi .643E+oi ,907E+oo

,167 E+02 .227 E+01 .644 E+01 ,589 E+O0

; BW—.FL 200-% BW

DECEMBFR

1966 WENZEL: CAPACITANCE MATRIX TRANSFORMATIONS 645

TABLE III

SELFA~D Mu’ruALsTA’rIc CAPACITORS FOR CHEBYSHEV INTERDIGITAL FILTERS WITH OPE~-CIRCUITED‘TERMINATING LINES m =3, n= 3, Nf = 6 SECTXONS

—

A

c = 0.01 DB VSWRM= 1,10:1

%BW <c” K’, -k’, &’, %“%’4

30,035,040.045,0

50,055.060,065.070.07!f. o

80,085.090,095.0

100.0110,0120,0130.0140.0150.0

,232E+01,273 E+01

.315E+01

,358E+01,403 E+oi,449E+o I,497E+01,548E+o I

,601E+01,656E+01,716E+01,779E+01,847E+01

,920 E+01.999 E+Oi,118E+02,140E+02

,168 E+02,205E+02

,257E+02

.368 E+02 ,678 E+01 ,314E+OZ .640E*01

.306E+02 .679E+01 .’253E+02 .641!2+01

.260 E+02 .679E*01 .208E+02 .642 E+01

.224E+02 .680 E+01 .174E+02 .644E+ol

.195E+02 .681 E+01 .146E*02 .645E+01

.171 E+02 .682E+01 .124E+02 .647E+01

.151 E+02 .683E+01 .106E*O2 .649E+01

.135E+02 .684E*01 ,906E+01 .651 E+Oi●120 E+02 .685 E+01 .778E+01 .654 E+01.108E+O2 .687E+01 ,670 E+01 ,656E+01.971 E+o I .688 E+o I ,577E+01 .659E+01

,875E+01 .690 E+01 ,497E*01 .662E+OI,789E+01 .691 E+OI ,427E+OI ,664 E+01.713E+01 .693E+01 .367E+01 ,667E+01

,644 E+01 .694E+OI .314 E+01 ,671F+01.525E+01 .698 E+01 .?29E+Oi .677E+01,426E+01 .701 E+01 .163E+01 ,684 E+01.342E+01 .704E+01 .i13E+01 .690E+01.270E+01 .708 E+01. .751E*O0 .697E+01.208E+01 .711 E+01 .473E+O0 .703E+01

FH— —

B

r = O.IODB VSWR-M= 1.36:1

30,035.040.045.050.055.060.065. o70,0

75.080.085.0

90.095.0

100.0110.0120.0

130.0140,0

150.0

,155E+OI ,3 S6E+02 .624E+01,182 E+01 .322 E+02 .624 E+01.210 E+o1 .274E+02 .624E+01

,239 E+01 .2if6E+02 .625 E+OI,268E+01 ,206E+02 .625E+01.299E+01 .182E+02 .626E+OI.331E+01 .16i E+02 ,626E+OI.364E+01 .144 E+02 .627E+01.399 E+01 .129 E+02 .627 F+01

.435E+o I .116E+02 .628E+Oi

.474E+01 .104E+O2 .629E.01

,515E+01 .943E+01 .630F+01,559E+01 ,854E+01 .630E+o I,606 E+01 .773 E+01 .631 E+01

,657E+o I .701 E+01 .632E+o I

.773E+01 .575E+01 .634E+01,914 E+01 ,469 E+01 .636 E+01

,109E+O2 .3 BoE+01 .63 BE+01,132E+02 .302E+01 .640E+01.164E+02 ,235E+01 .642E+01

.334E+02

.271E+02

.225E+02

.188E+02

.160 E*02

.136E+02

.117E+02

.101 E+o2

. 874 E+o I

.758E+01

.658E+OI

.571E+01

.496 E+o I

. 430 E*01

.372E+01

.276E*01

.201E*01

.143E+01,975E*o0

,631E+o0

+%BW

,601 E+01,601 F+01.602F+01,603E+01.604E+01.605E+01.606E+01,607 E+01.608E+01

,609E+01,61 OE*O1,612 F+01

.613 E+01,615E+01

,617 F+01

.620 E+01

.624 F+01

,628E+01,632E+01,63 bE+ol

FL 200- % BW

‘s=lmml[almRL

C4“,=1+2 “3=1+x%—

RS C2 C2 C2 n2. ,; ,22 ,; WHERE

~ C2+ C3 C2“2=1+ —. —

C4 .4 n]

Cl = 1/35.2 ohms C2 = 1/11.2 ohm,

G

AI :1 t? l+--+..= 11.0/( .2. 7.19~ ‘c4r11.01fi

‘S=’UIRL=l

(.) Normalized Static Capacitance Values Obtninei Fr.m Table I. A

CO’ ll;/fi :2= l;58~ ‘4= ‘“{f ,

,~ .3=258/~73 :=3””’

(b) Resultant Network After One Tr.nsfmnati.n of the C.p.citanu Netw.rk

jufi:=’’ii’’:;:q:p(c) Network wiih Largest Possible Trumsfwner Ratio

Fig. 7. Coaxial filter-transformer design.

50 ohmsZ = 30 ohms 16.3 ohms

Lle

A, B, c, H,

(u) S.PI... Pr.t.type Nelw.rk N.rm.lfzed t. 50 ohms

B ,21 = 35.2 ohm, ,22 = 30 ohms

(b) Schematic Cr.ss-Secti.n.l Drawing

Fig. 8. Coaxial filter-transformer schematic diagram.

646 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

I.D. = .164” L

l.D. = .l13”~

.0

BRM

.250”

.050”

(1.4” DEEP)

I.D. = .238” ~

TEFLON TO

BOTTOM OF HOLE

I I. D.= .147”t .001’” (1,400”/

JJTEFLON TO

.001” BOTTOM OF HOLE

II I.113”

II I ‘1II III I

.050”

II [ 711111

1

1.050”

1~~~ .200” .166”

II I Ll~j

-4 ?- ~50,,

BRM

t

BRASS RING

O.D. .237”

z0

.loo”i .005”

4

C;.,:::.oo,! ;

‘+

1-------- ----------------------------------------- .

-f

~1.7Do” f .003”-s

1 I-------—--------—-----------------------

DEEP)

e J-.

----- ______ ______

-4=-.100” :40:;;“003”+:z.100” * .005” +1

G.

1.0. .166”

Fig. 9, Assembly drawing of 3:1 bandwidth filter-transformer.

6.0T

5.0

p%. ~

THEORETICAL VSWR

~ 4,0 MEASUREDVSWR

~> 3,0 -. . ..~-—= —- L

i.D

1.0 ! I , , i

1.0 1.5 2,0i

2.5FREQ (GHz)

30

(.) lnPUI V5WR .t L.. Impedance Terminal With A 50. ohm GenW.TfOr

1.51,4

~s7/%,_., ;~

CONVERTED THEORETICAL VSWRMAX. VSWR = 1,24

g T.3– —-,———-—- ———_’w!u____ _—— -S ;:; CON VERTEO MFASUREO VSWR

~--1.0

--—--+ ~

1.0 1.5 2,0 2,5 30FREQ (GHz)

(b) Cwnputed Input VSWR et LOW Impedance TermInol With A 16.3..hm Generotor

Fig. 10. Theoretical and measured performance of 3:1 bandwidthfilter-transformer.

700” ~ .005”

Cjb J.562” ~ .001”

Fig. 11. Four-section coaxial filter-transformer.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TEcHNIQUES, VOL. MTT-14, NO. 12,

CONCLUSIONS

The description of TENI propagation on an array of

parallel coupled lines in terms of the {static capacitance

matrix allows a unified treatment of many heretofore

seemingly unrelated network configurations. The use

of the capacitance matrix transformation provides a

simple method, devoid of complicated mathematics, for

investigating possible equivalent circuit structures and

obtaining new network forms. Furthermore, the tech-

niques described provide considerable physical insight

into the meaning of equivalent TEIM networks and

should be a very useful tool, both computationally and

conceptually, to the network designer.

[1]

[2]

[3]

[4]

I?EFERENCES

R. J. Wenzel, “Exact theory of interdigital band-pass filters andrelated coupled structures, ” IEEE T~ans. on Microwave Theoryand Techniques, vol. MTT-13, pp. 559–575, September 1965.S. 0. Rice, “Steady-state solutions of transmission line equa-tions, ” Bell Sys. Tech. J., vol. 20, pp. 131-178, April 1941.D. C. Youla, ‘[Au introduction to coupled-line network theory, ”hIIRI, Polytechnic Institute of Brooklyn, Brooklyn, N. Y.,Rept. 96G961, 1961.E. Ott, “A network approach to the design of multiline 2N-portdirectional couriers. ” Polytechnic Institute of BrooklvmBrooklyn, N. Y:, Rept. PIBMRI-1236-64; Air Research De:velopment Command, Rome Air Development Center, Griffiss

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

DECEMBER, 1966 647

AFB, Rome, N. Y,, Contract AF 30(602)-2868, Tech. Rept.RADC-TR-65-41, April 1965.W. J. Getsinger, “Coupled rectangular bars between pa,rallelplates, ” IRE Trans. on Microwave Theory and Techniques, vol.MTT-10, pp. 65-72, January 1962.E. G. Cristal, “Coupled circulator cylindrical rods betweenparallel ground planes, ” IEEE Trans. on Microwave Theory andTe.7zniques, vol. MTT-12, pp. 428-439? July 1964.R. W. Beattv and D. M. Kerns. “Relatlonshim between differentkinds of net~ork parameters, not assuming ;eciprc,city or eqtsal-ity of the waveguide or transmission line characteristic im-penances, ” P~oc. IEEE (Corresfiondence), vol. 52!, p. 84, lan-.-. .Gary 1964.E. A. Guillemin, Synthesis of Passive Networks,, New York:Wilev. 1957.

--–J ,

R. M. Fano, L. J. Chu, and R. B. Adler, Electromagnetic FYelds,Energy and Forces. New York: Wdey, 1960.E. M. T. Jones and J. T. Bolljahn, “Coupled-strip-transmi ssion-line filters and directional couplers, ” IRE l“mm. on MicmzvaveTheory and Techniques, vol. MTT-4, pp. 75–81, April 19S6.S. B. Cohn, “The re-entrant cross section and wide-band .3-dBhybrid couplers,” IEEE Trans. on Microwave Theory and Tech-niques, VO1. MTT-11, pp. 254–25% Tlllv 196.3R. J. Wenzel and M. C. Hormicro~

---7 ., --, -----

rton, “Exact design techniques forwave TEM filters, ” Final Rept., U. S. Army Electronics

Lab.. Fort Monmouth. N. T.. 00399(E). Bendix Research Labs..Southfield, Mich., Ap~l 19”65. ‘ ‘‘The Microwave Engineers Handbook and Buyer’s Guide. Brook-Iine, Mass.: Horizon House-Microwave, Jnc., 1966, p. 91.M. C. Horton and R. 1. Wenzel, ‘[General theorv and desire ofoptimum quarter-wav; TEM filters, ” IEEE T;ans. on lJ&ro-

~~~ Theory and Techniques, vol. MTT-13, pp. 316–327, May. ../”.

L. Young, “Tables for cascaded homogeneous quarter-wavetransformers, ” IRE Trans. on Microwave Theory and Techniques,vol. MTT-9, pp. 233–237, April 1959.

The Design and Construction of Broadband,

High~Directivity, 90~Degree Couplers Using

Nonuniform Line Techniques

c. p. TRESSELT, MEMBER, IEEE

Abstract—It is possible, at present, to obtain multioctave band-width in symmetrical couplers that employ cascaded quarter-wave-length sections of uniformly coupled line. However, the physical

junctions between the various sections contribute reactive dlscon-tinuities, which significantly degrade coupler directivity. This paper

describes a coupler design employing a continuously tapered coupling

coefficient that helps to circumvent the dlrectivity problem.Two classes of couplers have been investigated, including one

which provides optimum equal-ripple performance. Synthesis has

been performed with the aid of both digital and Fourier integral

computers. Somewhat tighter center coupling is required in thetapered design to produce bandwidth-mean-coupling-level per-

formance comparable to stepped-coupling design.Experimental data is presented on several models constructed in

three-layer polyolefin stripline.

Manuscript received June 29, 1966; revised August 8, 1966.The author is with The Bendix Corporation, Research Labora-

tories Division, Southfield, Mich.

1. INTRODUCTION

I

T IS POSSIBLE to obtain multioctave bandwidth

from a single coupler that employs several cas-

caded quarter-wavelength sections of uniformly

coupled line. The use of a symmetric structure guaran-

tees 90° relative phase lead of the coupled port with

respect to the transmitted port at all frequencies. ‘Cris-

tal and l“oung [1] present tables of coupling coefficients

required to produce equal-ripple response for the ttlird-

through ninth-order designs, and include a ,comprehen-

sive bibliography covering the historic development of

this form of coupler. Tight coupling normally is required

in at least one of the sections of a broadband design.

This condition has been alleviated to some extent by the

application of tandem interconnection [2].